L(s) = 1 | + (0.866 − 0.5i)3-s + (0.866 + 0.5i)7-s + (0.499 − 0.866i)9-s + 1.73i·13-s + (−0.866 − 0.5i)19-s + 0.999·21-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (−0.866 − 1.5i)31-s + (−1.5 − 0.866i)37-s + (0.866 + 1.49i)39-s − i·43-s + (0.499 + 0.866i)49-s − 0.999·57-s + (0.866 − 0.499i)63-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)3-s + (0.866 + 0.5i)7-s + (0.499 − 0.866i)9-s + 1.73i·13-s + (−0.866 − 0.5i)19-s + 0.999·21-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (−0.866 − 1.5i)31-s + (−1.5 − 0.866i)37-s + (0.866 + 1.49i)39-s − i·43-s + (0.499 + 0.866i)49-s − 0.999·57-s + (0.866 − 0.499i)63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.539140871\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.539140871\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - 1.73iT - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + 2T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.305375519762864375978932383853, −9.040180286002621190107320790153, −8.281563512959628639286162862746, −7.31776831144826739565169662696, −6.77261580839735768819170297878, −5.66656599874951787414939501177, −4.52946621521874778386516169117, −3.73619799099786836484352427627, −2.32637257126767035231021023640, −1.71936963225688645783096004183,
1.56163922652239484991610100398, 2.84874015659539083664884360037, 3.71094013236821201846506839146, 4.73069243349022429596076097372, 5.38537914834218028578613808840, 6.71263271333190800662654143049, 7.70797342041794686363990231207, 8.257949204544708563433343436772, 8.809403759642098163147127063327, 10.00698876290157562610644250422